CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A nonfree action can have $|X/G|\ne|X|/|G|$

Statement refuted

False claim. For every action of a finite group $G$ on a finite set $X$ , the number of orbits is $|X|/|G|$ .

Facts & Assumptions

Given: The trivial action of $G=\mathbb Z/2$ on the singleton $X=\{*\}$ .

[L1]

Orbit counting uses the fixed-point sum $|G|\,|X/G|=\sum_{g\in G}|X^g|$ (Cauchy-Frobenius orbit counting: $|G|\,|X/G|=\sum_{g\in G}|X^g|$ for a finite group action).

[L2]

The trivial $\mathbb Z/2$ -action on a singleton is transitive and nonfaithful (The trivial action of $\mathbb Z/2$ on a singleton is transitive but not faithful).

Counterexample

technique · direct

1.1

By [L2], the singleton is one orbit, so $|X/G|=1$ .

2.1

Here $|X|=1$ and $|G|=2$ , so there is no natural number $q$ with $|X|=|G|q$ ; in particular the orbit count is not obtained by dividing $|X|$ by $|G|$ .

step 1.1L2algebra

3.1

Both elements of $G$ fix the unique point, so [L1] correctly gives $2\cdot1=1+1$ . This verifies the orbit-counting identity while refuting the naive division rule.

step 1.1step 2.1L1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 56 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

T. W. Judson, Abstract Algebra: Theory and Applications, 14.3 (standard reference, not scraped)

A nonfree action can have ∣X/G∣≠∣X∣/∣G∣|X/G|\ne|X|/|G|∣X/G∣=∣X∣/∣G∣